Question

If ${\displaystyle \overline{a}}$, ${\displaystyle \overline{b}}$ and ${\displaystyle \overline{c}}$ are three non-coplanar vectors then ${\displaystyle (\overline{a}+\overline{b}+\overline{c}).[(\overline{a}+\overline{b})\times (\overline{a}+\overline{c})]}$ = ______

विकल्प

• 0

• ${\displaystyle [\overline{a} \overline{b} \overline{c}]}$

• ${\displaystyle -[\overline{a} \overline{b} \overline{c}]}$

• 2${\displaystyle [\overline{a} \overline{b} \overline{c}]}$

Answer 1

${\displaystyle -[\overline{a} \overline{b} \overline{c}]}$

Explanation:

${\displaystyle [\overline{a}+\overline{b}+\overline{c}].[(\overline{a}+\overline{b})\times (\overline{a}+\overline{c})]}$

= ${\displaystyle (\overline{a}+\overline{b}).[(\overline{a}+\overline{b})\times (\overline{a}+\overline{c})]+\overline{c}.[(\overline{a}+\overline{b})\times (\overline{a}+\overline{c})]}$

= 0 + ${\displaystyle [\overline{c} \overline{a}+\overline{b} \overline{a}+\overline{c}]}$

= ${\displaystyle [\overline{c} \overline{a} \overline{a}+\overline{c}]+[\overline{c} \overline{b} \overline{a}+\overline{c}]}$

= ${\displaystyle [\overline{c} \overline{a} \overline{a}]+[\overline{c} \overline{a} \overline{c}]+[\overline{c} \overline{b} \overline{a}]+[\overline{c} \overline{b} \overline{c}]}$

= ${\displaystyle 0+0+[\overline{c} \overline{b} \overline{a}]+0=-[\overline{a} \overline{b} \overline{c}]}$